Mahalanobis distance invariant against further elements/individuals?

Assume following: There are ten individuals and each is represented by two properties (Size and Gender). Now I measure the distance between two individuals A and B via the Mahalanobis distance:

$$d(x, y)=\sqrt{\left(\bar{x}-\bar{y}\right)^{T}S^{-1}\left(\bar{x}-\bar{y}\right)}$$

and get a certain distance. After that I add a few more individuals to the pool and I measure again the distance of the same both individuals A and B as before: Will it be different then (because of the covariance matrix)?

edit: I just tried it out and it's different / the distances between individuals differ with the pool they are part of it.

• Your notation is a bit unclear. What do you mean by the, $x$, $y$, $\bar{x}$ and $\bar{y}$? Also, is $S$ a given covariance matrix, or do you estimate it from the sample? What is the multi-dimensional variable in this problem? – Sextus Empiricus Dec 22 '17 at 15:23
• x and y are the individuals (e.g. x is individual A and y is individual B). I estimate it from the sample/pool? Is there a difference between sample and pool? In this example the variable would be A = x = (Size_{A}, Gender_{A}) and B = y = (Size_{B}, Gender_{B}). – Ben Dec 22 '17 at 15:32
• That is still not very complete. What are $\bar{x}$ and $\bar{y}$? How do you estimate $S$? New questions that arise are: 1 what is the meaning of your distance in the case of categorical variables (gender)? 2 how do you do the multiplication of the different types of variables? 3 What is the meaning of this distance operation, why do you do it? – Sextus Empiricus Dec 22 '17 at 17:44
• x and y are the vectors of the individuals comprising their properties. How I estimate S? Do you mean how I program it? I've done it via stackoverflow.com/a/15142446/6761328 . Relative to your questions: 1) I assume there is no meaning 2) Guess this is connected to the implementation of the calculation I've cited before(?) 3) I want to measure how similar the individuals are with respect to each other. My motivation is to find pairings (of man/woman) which are as similar as possible. – Ben Dec 24 '17 at 9:23

It is only because you have a small sample size and thus a "poor" estimation of $S$, which may be greatly influenced by the new individuals. Try it out with $n < 10000$ and, as long as the new individuals is not a high impact outlier, you should obtain the approximately the same distance between individuals A and B when adding the $10001$th subject.

EDIT: To illustrate the answer below, try this code in R :

> library(MASS)
> S = matrix(c(1, .5,.5,1),2,)
> data = mvrnorm(n=1000,mu=rep(0,2),Sigma=S)
> cov(data)
[,1]      [,2]
[1,] 1.0301079 0.4881467
[2,] 0.4881467 1.0292714
> round(mahalanobis(data[1:10,],center=mean(data),cov=cov(data)),4)
[1] 1.1544 2.5189 0.2999 0.6342 2.4010 2.2885 1.2370 1.6227 1.1656
[10] 1.3078
> new = mvrnorm(n=1,mu=rep(0,2),Sigma=S)
> data2= rbind(data,new)
> round(mahalanobis(data2[1:10,],center=mean(data),cov=cov(data)),4)
[1] 1.1544 2.5189 0.2999 0.6342 2.4010 2.2885 1.2370 1.6227 1.1656
[10] 1.3078


You can see that the 10 persons kept the same distance. We can even change the new $n$ individuals to 100 and mahalanobis distance won't change much

> library(MASS)
> S = matrix(c(1, .5,.5,1),2,)
> data = mvrnorm(n=1000,mu=rep(0,2),Sigma=S)
> cov(data)
[,1]      [,2]
[1,] 0.9506534 0.4780387
[2,] 0.4780387 0.9835325
> round(mahalanobis(data[1:10,],center=mean(data),cov=cov(data)),4)
[1] 3.5866 7.0324 1.1189 2.0611 4.5880 0.1649 3.2801 4.4306 2.4925
[10] 1.9930
> new = mvrnorm(n=100,mu=rep(0,2),Sigma=S)
> data2= rbind(data,new)
> round(mahalanobis(data2[1:10,],center=mean(data),cov=cov(data)),4)
[1] 3.5866 7.0324 1.1189 2.0611 4.5880 0.1649 3.2801 4.4306 2.4925
[10] 1.9930


Considering all participants :

> library(MASS)
> S = matrix(c(1, .5,.5,1),2,)
> data = mvrnorm(n=1000,mu=rep(0,2),Sigma=S)
> cov(data)
[,1]      [,2]
[1,] 0.9580746 0.4467521
[2,] 0.4467521 0.9838970